In the wiki, I found this definition for the argument: http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitions However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for argument:[tex]\arg(z) = \ln(\sqrt[2 i]{z \div \bar{z} }) = \frac{ln(z) - ln(\bar{z})}{2 i}[/tex]I don't understand why exist 2 defitions for the argument and how those 2 defitions are related.
This gives the inverse of [itex]\mathrm{cis}\,\theta = \cos \theta + i \sin \theta = e^{i\theta}[/itex]. It is not a definition of the argument, but reflects the fact that if [itex]z = e^{i\theta}[/itex] then [tex] -i \log e^{i\theta} = -i(i \theta) = \theta = \arg z. [/tex] It doesn't give [itex]\arg z[/itex] if [itex]|z| = R \neq 1[/itex]: [tex] -i \log (Re^{i\theta}) = -i \log R + \theta \neq \arg z [/tex] This gives [itex]\arg z[/itex] for any [itex]z \neq 0[/itex] (if you choose the correct branch of [itex]z^{1/(2i)}[/itex]).
There is almost always an alternative way of expressing the same mathematical argument, with a little imagination. It's not always obvious.